

A250207


The number of quartic terms in the multiplicative group modulo n.


6



1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 4, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 2, 5, 4, 3, 3, 9, 9, 3, 1, 10, 3, 21, 5, 3, 11, 23, 1, 21, 5, 4, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 4, 3, 5, 33, 4, 11, 3, 35, 3, 18, 9, 5, 9, 15, 3, 39, 1
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OFFSET

1,7


COMMENTS

In the character table of the multiplicative group modulo n there are phi(n) different characters. [This is made explicit for example by the number of rows in arXiv:1008.2547.] The set of the fourth powers of the characters in all representations has some cardinality, which defines the sequence.


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000
R. J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, (2017).
R. J. Mathar, Table of Dirichlet Lseries and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010.
Wikipedia, Dirichlet character


FORMULA

a(n) = A000010(n)/A073103(n).
Multiplicative with a(2^e) = 1 for e<=3; a(2^e) = 2^(e4) for e>=4; a(p^e) = p^(e1)*(p1)/4 for e>=1 and p == 1 (mod 4); a(p^e) = p^(e1)*(p1)/2 for e>=1 and p == 3 (mod 4). (Derived from A073103.)  R. J. Mathar, Oct 13 2017


EXAMPLE

For n <= 6, the set of all characters in all representations consists of a subset of +1, 1, +i or i. Their fourth powers are all +1, a single value, so a(n)=1 then.
For n=7, the set of characters is 1, 1, +1/2 + sqrt(3)*i/2, so their fourth powers are 1 or 1/2 + sqrt(3)*i/2, which are three different values, so a(7)=3.
For n=11, the fourth powers of the characters may be 1, exp(+2*i*Pi/5) or exp(+4*i*Pi/5), which are 5 different values.


MAPLE

A250207 := proc(n)
numtheory[phi](n)/A073103(n) ;
end proc:


MATHEMATICA

a[n_] := EulerPhi[n]/Count[Range[0, n1]^4  1, k_ /; Divisible[k, n]];
Array[a, 80] (* JeanFrançois Alcover, Nov 20 2017 *)


PROG

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]==2, 2^max(0, f[i, 2]4), f[i, 1]^(f[i, 2]1)*(f[i, 1]1)/if(f[i, 1]%4==1, 4, 2))) \\ Charles R Greathouse IV, Mar 02 2015


CROSSREFS

Cf. A046073, A087692, A052273, A293482  A293485.
Sequence in context: A213621 A053575 A293485 * A216319 A309425 A218355
Adjacent sequences: A250204 A250205 A250206 * A250208 A250209 A250210


KEYWORD

easy,nonn,mult


AUTHOR

R. J. Mathar, Mar 02 2015


STATUS

approved



